TITLE: D-modules and Algebraic Solutions of PDEs 
QUESTION [20 upvotes]: I am not certain if this is a complete question and I fear it might be shot down. Anyway, I try to pose it. My question is in connection to  using D-modules to  study PDEs (and systems of PDEs). When I was doing a perusal on "A primer of algebraic D-modules by  S. C. Coutinho" the justification on the importance of D-modules; they provide an algebraic tool towards the solution of differential equations. This is the story I always hear!  Do someone have a reference or more information about D-modules and algebraically solution of PDEs ?.

REPLY [12 votes]: My guess is that this is a reference to Bernshtein's proof using the theory of D-modules that every constant coefficent partial differential operator $D$ has a fundamental solution, i.e. there is a distributional solution to the PDE $Du = \delta$ where $\delta$ is the Dirac distribution.  This was an important theorem in analysis due to Malgrange and Ehrenpreis in the 1950's, and I think it came as a bit of a surprise that it can be done purely algebraically - all of the analytic input is encapsulated by a few basic facts about distributions.  Everyone knows that the fundamental theorem of algebra is proved in analysis class; maybe this means the fundamental theorem of analysis will one day be proved in algebra class!
Here is a link to Bernshtein's paper: http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/bernstein-mod-dif-FAN.pdf.  This was written in the 70's, and I think since then the argument has been cleaned up and made even more algebraic.